Where would this be on a range? ($0$ to $100\%$) Algebra

Question

Ok, we have two expressions:

$30x + 1500$
$19x + 9000$

The first thing to note is, the first expression is $100\%$ and the second expression is $0\%$.

The second thing is to note in all of this is, $x$ is infinite in these expressions.

Ok, my main question is where would $23x + 15590$ go in the range as a percent? and how do you prove this? I don't know how but I'll accept any answer that makes sense. Bearing in mind $x$ is infinite, how?

Thank you though, I look forward to your answers.

EDIT: Hello guys, I apologize if that didn't make any sense. I meant in terms of this: link

Answer 1

I guess you mean three values linearly dependent on $x$, say $f(x), g(x), h(x),$ the third one falling between the former two for $x$ big enough; and you are asking about an asymptotic value of the proportion $$\frac{h-f}{g-f}$$ as $x$ grows to infinity.

That is $$\begin{align} \lim_{x\to\infty}\frac{(23x+15590)-(19x+9000)}{(30x+1500)-(19x+9000)} & = \lim_{x\to\infty}\frac{4x+6590}{11x-7500} \\ & = \lim_{x\to\infty}\frac{4+6590/x}{11-7500/x} \\ & = \frac{4+0}{11-0} \\ & = \frac 4{11} \\ & \approx 0,363636 \\ & \approx 36.4\% \end{align}$$

For example, here are some values of the fraction defined above:
for $x=10^3\ \,$ it is $\approx 3.025714286$,
for $x=10^4\ \,$ it is $\approx 0.454536586$,
for $x=10^8\ \,$ it is $\approx 0.363644834$ and
for $x=10^{10}$ it is $\approx 0.363636448$.

But please note that the limit value as $x$ grows to infinity is not the same as a 'proportion of infinite values'! The former exists and can be calculated, the latter simply does not exist, is undefined.

Answer 2

Your objects are algebraic expressions. Among the possible interpretations are linear functions and straight lines in the plane. Interpreting them as linear functions would look like this: $$ f(x) = 30x + 1500 \\ g(x) = 19x + 9000 \\ h(x) = 23x + 15590 $$

The only thing that makes remotely sense to me is that you have some interpolation problem. $$ \phi(1) = f \\ \phi(0) = g \\ \phi(\lambda) = h $$ where $\phi$ is a map that maps a parameter $\lambda$ to some linear function. E.g. $$ \phi(\lambda) = \lambda f + (1-\lambda) g $$ would be such an interpolation. Alas your $h$ is probably not within the reachable functions of the above $\phi$.

Let us check it: $$ \phi(\lambda) = \lambda(30x + 1500) + (1-\lambda)(19x +9000) \\ = (11 \lambda + 19) x + (9000 - 7500 \lambda) $$ comparison with $h$ gives the conditions $$ 11 \lambda + 19 = 23 \\ 9000 - 7500 \lambda = 15590 $$ or $$ \lambda = 4/11 > 0 \\ \lambda = -6590/7500 < 0 $$ so there is no $\lambda \in \mathbb{R}$ which would result in $\phi(\lambda) = h$.